>>Geometry: egyptian triangle. Complete Lessons
Lesson topic
Lesson Objectives
- Get acquainted with new definitions and recall some already studied.
- Deepen knowledge of geometry, study the history of origin.
- To consolidate the theoretical knowledge of students about triangles in practical activities.
- To introduce students to the Egyptian triangle and its application in construction.
- Learn to apply the properties of shapes in solving problems.
- Developing - to develop students' attention, perseverance, perseverance, logical thinking, mathematical speech.
- Educational - through a lesson, to cultivate an attentive attitude towards each other, to instill the ability to listen to comrades, mutual assistance, independence.
Lesson objectives
- Check students' ability to solve problems.
Lesson Plan
- Introduction.
- Good to remember.
- Triangle.
introduction
Did the ancient Egyptians know mathematics and geometry? They not only knew, but also constantly used it to create architectural masterpieces and even ... in the annual marking of fields on which water destroyed all the boundaries during a flood. There was even a special service of land surveyors who quickly restored the boundaries of the fields with the help of geometric techniques when the water subsided.
It is not yet known what we will call our young generation, which grows up on computers that allow us not to memorize the multiplication table and not to perform other elementary mathematical calculations or geometric constructions in our minds. Maybe human robots or cyborgs. The Greeks, on the other hand, called those who could not prove a simple theorem without outside help, profane. Therefore, it is not surprising that the very theorem, which was widely used in applied sciences, including for marking fields or building pyramids, was called by the ancient Greeks the “bridge of donkeys”. And they knew Egyptian mathematics very well.
Good to remember
Triangle
Triangle rectilinear, part of the plane bounded by three line segments (sides of the Triangle (in geometry)), having in pairs one common end (vertices of the Triangle (in geometry)). A triangle in which the lengths of all sides are equal is called equilateral, or right, Triangle with two equal sides - isosceles. The triangle is called acute-angled if all its angles are acute; rectangular- if one of its corners is right; obtuse- if one of its corners is obtuse. A triangle (in geometry) cannot have more than one right or obtuse angle, since the sum of all three angles is equal to two right angles (180° or, in radians, p). The area of a Triangle (in geometry) is equal to ah/2, where a is any of the sides of the Triangle taken as its base, and h is the corresponding height. The sides of the Triangle are subject to the condition: the length of each of them is less than the sum and greater than the difference in the lengths of the other two sides.
Triangle- the simplest polygon, having 3 vertices (corners) and 3 sides; a part of a plane bounded by three points and three line segments connecting these points in pairs.
- Three points in space that do not lie on one straight line correspond to one and only one plane.
- Any polygon can be divided into triangles - this process is called triangulation.
- There is a section of mathematics entirely devoted to the study of the patterns of triangles - Trigonometry.
Triangle types
By the type of angles
Since the sum of the angles of a triangle is 180°, at least two angles in a triangle must be acute (less than 90°). There are the following types of triangles:
- If all angles of a triangle are acute, then the triangle is called acute;
- If one of the angles of the triangle is obtuse (greater than 90°), then the triangle is called obtuse;
- If one of the angles of a triangle is right (equal to 90°), then the triangle is called a right triangle. The two sides forming a right angle are called the legs, and the side opposite the right angle is called the hypotenuse.
By the number of equal sides
- A triangle is called scalene if the lengths of three sides are pairwise different.
- An isosceles triangle is one in which two sides are equal. These sides are called the sides, the third side is called the base. In an isosceles triangle, the angles at the base are equal. The height, median and bisector of an isosceles triangle, lowered to the base, are the same.
- An equilateral triangle is one in which all three sides are equal. In an equilateral triangle, all angles are equal to 60 °, and the centers of the inscribed and circumscribed circles coincide.
- a right triangle with an aspect ratio of 3:4:5. The sum of these numbers (3+4+5=12) has been used since ancient times as a unit of multiplicity when constructing right angles using a rope marked with knots at 3/12 and 7/12 of its length. The Egyptian triangle was used in the architecture of the Middle Ages to build proportionality schemes.
So where do you start? Is it from this: 3 + 5 = 8. and the number 4 is half of the number 8. Stop! The numbers 3, 5, 8... Don't they look very familiar? Well, of course, they are directly related to the golden ratio and are included in the so-called "golden row": 1, 1, 2, 3, 5, 8, 13, 21
... In this series, each subsequent term is equal to the sum of the two previous ones: 1 + 1= 2. 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8
and so on. It turns out that the Egyptian triangle is related to the golden ratio? And did the ancient Egyptians know what they were dealing with? But let's not jump to conclusions. It is necessary to find out the details more precisely.
Expression " golden ratio”, according to some, was first introduced in the 15th century Leonardo da Vinci
. But the “golden row” itself became known in 1202, when it was first published in his “Book of Accounts” by an Italian mathematician Leonardo of Pisa
. Nicknamed Fibonacci. However, almost two thousand years before them, the golden ratio was known Pythagoras and his students. True, it was called differently, as "division in the middle and extreme ratio." And here is the Egyptian triangle with its The "golden ratio" was known back in those distant times when the pyramids were built in Egypt when Atlantis flourished.
To prove the Egyptian triangle theorem, it is necessary to use a straight line segment of known length A-A1 (Fig.). It will serve as a scale, a unit of measurement, and will allow you to determine the length of all sides of the triangle. Three segments A-A1 are equal in length to the smallest of the sides of the triangle BC, in which the ratio is 3. And four segments A-A1 are equal in length to the second side, in which the ratio is expressed by the number 4. And, finally, the length of the third side is equal to five segments A -A1. And then, as they say, a matter of technology. On paper, draw a segment BC, which is the smallest side of the triangle. Then, from point B with a radius equal to the segment with a ratio of 5, we draw an arc of a circle with a compass, and from point C, an arc of a circle with a radius equal to the length of the segment with a ratio of 4. If now the intersection point of the arcs is connected by lines with points B and C, then we get a right triangle aspect ratio 3:4:5.
Q.E.D.
The Egyptian triangle was used in the architecture of the Middle Ages to build proportionality schemes and to build right angles by land surveyors and architects. The Egyptian triangle is the simplest (and first known) of the Heronian triangles - triangles with integer sides and areas.
Egyptian triangle - a mystery of antiquity
Each of you knows that Pythagoras was a great mathematician who made an invaluable contribution to the development of algebra and geometry, but he gained even more fame thanks to his theorem.
And Pythagoras discovered the Egyptian triangle theorem at the time when he happened to visit Egypt. While in this country, the scientist was fascinated by the splendor and beauty of the pyramids. Perhaps this was precisely the impetus that made him think that some definite pattern was clearly traced in the forms of the pyramids.
Discovery history
The name of the Egyptian triangle was due to the Hellenes and Pythagoras, who were frequent guests in Egypt. And it happened around the 7th-5th centuries BC. e.
The famous pyramid of Cheops, in fact, is a rectangular polygon, but the sacred Egyptian triangle is considered to be the pyramid of Khafre.
The inhabitants of Egypt compared the nature of the Egyptian triangle, as Plutarch wrote, with the family hearth. In their interpretations, one could hear that in this geometric figure, its vertical leg symbolized a man, the base of the figure belonged to the feminine, and the hypotenuse of the pyramid was assigned the role of a child.
And already from the topic studied, you are well aware that the aspect ratio of this figure is 3:4:5 and, therefore, that this leads us to the Pythagorean theorem, since 32 + 42 = 52.
And if we take into account that the Egyptian triangle lies at the base of the Khafre pyramid, then we can conclude that the people of the ancient world knew the famous theorem long before it was formulated by Pythagoras.
The main feature of the Egyptian triangle, most likely, was its peculiar ratio of sides, which was the first and simplest of the Heronian triangles, since both the sides and its area had integers.
Features of the Egyptian triangle
And now let's take a closer look at the distinctive features of the Egyptian triangle:
• Firstly, as we have already said, all its sides and area consist of integers;
• Secondly, by the Pythagorean theorem we know that the sum of the squares of the legs is equal to the square of the hypotenuse;
• Thirdly, with the help of such a triangle it is possible to measure right angles in space, which is very convenient and necessary in the construction of structures. And the convenience lies in the fact that we know that this triangle is a right triangle.
• Fourth, as we also already know that even if there are no appropriate measuring instruments, then this triangle can easily be built using a simple rope.
Application of the Egyptian triangle
In ancient times, the Egyptian triangle was very popular in architecture and construction. It was especially necessary if, to build right angle use rope or cord.
After all, it is known that laying a right angle in space is a rather difficult task, and therefore the enterprising Egyptians invented an interesting way to construct a right angle. For these purposes, they took a rope, on which twelve even parts were marked with knots, and then a triangle was folded from this rope, with sides that were equal to 3, 4 and 5 parts, and as a result, without any problems, they got a right triangle. Thanks to such an intricate tool, the Egyptians measured the land for agricultural work with great accuracy, built houses and pyramids.
This is how visiting Egypt and studying the features of the Egyptian pyramid prompted Pythagoras to discover his theorem, which, by the way, got into the Guinness Book of Records as the theorem that has the largest amount of evidence.
Reuleaux triangular wheels
Wheel- a round (as a rule), freely rotating or fixed on an axis disk, which allows the body placed on it to roll rather than slide. The wheel is widely used in various mechanisms and tools. Widely used for cargo transportation.
The wheel significantly reduces the energy costs for moving the load on a relatively flat surface. When using a wheel, work is done against the rolling friction force, which under artificial road conditions is significantly less than the sliding friction force. The wheels are solid (for example, the wheelset of a railway car) and consisting of quite a large number parts, for example, a car wheel includes a disk, a rim, a tire, sometimes a camera, mounting bolts, etc. Car tire wear is almost a solved problem (with properly set wheel angles). Modern tires travel over 100,000 km. An unresolved problem is tire wear on aircraft wheels. When a stationary wheel comes into contact with the concrete surface of the runway at a speed of several hundred kilometers per hour, tire wear is enormous.
- In July 2001, an innovative patent was obtained for the wheel with the following wording: "a round device used for transporting goods." This patent was issued to John Cao, a lawyer from Melbourne, who wanted to show the imperfection of the Australian patent law.
- The French company Michelin in 2009 developed a mass-produced Active Wheel with built-in electric motors that drive the wheel, spring, shock absorber and brake. Thus, these wheels make the following vehicle systems unnecessary: engine, clutch, gearbox, differential, drive and cardan shafts.
- In 1959, the American A. Sfredd received a patent for a square wheel. It easily walked through snow, sand, mud, overcame pits. Contrary to fears, the car on such wheels did not "limp" and developed a speed of up to 60 km / h.
Franz Relo(Franz Reuleaux, September 30, 1829 - August 20, 1905) - German mechanical engineer, lecturer at the Berlin Royal Academy of Technology, who later became its president. The first, in 1875, he developed and outlined the main provisions of the structure and kinematics of mechanisms; dealt with the problems of aesthetics of technical objects, industrial design, in his designs he gave great importance external forms of machines. Reuleaux is often called the father of kinematics.
Questions
- What is a triangle?
- Types of triangles?
- What is the peculiarity of the Egyptian triangle?
- Where is the Egyptian triangle used? > Mathematics Grade 8
Everyone who listened attentively to a geometry teacher at school is very familiar with what the Egyptian triangle is. It differs from other types of similar ones with an angle of 90 degrees by a special aspect ratio. When a person first hears the phrase "Egyptian triangle", pictures of majestic pyramids and pharaohs come to mind. And what does history say?
As is always the case, there are several theories regarding the name "Egyptian triangle". According to one of them, the well-known Pythagorean theorem saw the light precisely because of this figure. In 535 BC. Pythagoras, following the recommendation of Thales, went to Egypt in order to fill in some gaps in the knowledge of mathematics and astronomy. There he drew attention to the peculiarities of the work of Egyptian surveyors. They built in a very unusual way with a right angle, the sides of which were interconnected with one another in a 3-4-5 ratio. This mathematical series made it relatively easy to connect the squares of all three sides with one rule. This is how the famous theorem arose. And the Egyptian triangle is precisely the very figure that prompted Pythagoras to the most ingenious solution. According to other historical data, the Greeks gave the name to the figure: at that time they often visited Egypt, where they could be interested in the work of land surveyors. There is a possibility that, as is often the case with scientific discoveries, both stories happened at the same time, so it is impossible to say with certainty who came up with the name "Egyptian triangle" first. Its properties are amazing and, of course, are not limited to the aspect ratio alone. Its area and sides are represented by whole numbers. Due to this, the application of the Pythagorean theorem to it allows us to obtain integer numbers of squares of the hypotenuse and legs: 9-16-25. Of course, this could just be a coincidence. But how, then, to explain the fact that the Egyptians considered "their" triangle sacred? They believed in its interconnection with the entire universe.
After information about this unusual geometric figure became public, the world began searching for other similar triangles with integer sides. It was obvious that they existed. But the importance of the question was not just to perform mathematical calculations, but to test the "sacred" properties. The Egyptians, for all their unusualness, were never considered stupid - scientists still cannot explain exactly how the pyramids were built. And here, suddenly, a connection with Nature and the Universe was attributed to an ordinary figure. And, indeed, the found cuneiform contains indications of a similar triangle with a side whose size is described by a 15-digit number. Currently, the Egyptian triangle, the angles of which are 90 (right), 53 and 37 degrees, are found in completely unexpected places. For example, when studying the behavior of ordinary water molecules, it turned out that the change is accompanied by a restructuring of the spatial configuration of molecules, in which one can see ... the same Egyptian triangle. If we remember that it consists of three atoms, then we can talk about conditional three sides. Of course, we are not talking about the complete coincidence of the famous ratio, but the resulting numbers are very, very close to the desired ones. Is it because the Egyptians recognized their "3-4-5" triangle as a symbolic key to natural phenomena and the secrets of the universe? After all, water, as you know, is the basis of life. Without a doubt, it is still too early to put an end to the study of the famous Egyptian figure. Science never rushes to conclusions, seeking to prove its assumptions. And we can only wait and be surprised by the knowledge
Let's say we have a line to which we need to set a perpendicular, i.e. another line at an angle of 90 degrees relative to the first. Or we have an angle (for example, the corner of a room) and we need to check if it is equal to 90 degrees.
All this can be done with just a tape measure and a pencil.
There are two great things like the "Egyptian Triangle" and the Pythagorean theorem that will help us with this.
So, egyptian triangle is a right triangle with the ratio of all sides equal to 3:4:5 (leg 3: leg 4: hypotenuse 5).
The Egyptian triangle is directly related to the Pythagorean theorem - the sum of the squares of the legs is equal to the square of the hypotenuse (3*3 + 4*4 = 5*5).
How can this help us? Everything is very simple.
Task number 1. You need to draw a perpendicular to a straight line (for example, a line at 90 degrees to a wall).
Step 1. To do this, from point No. 1 (where our corner will be) you need to measure on this line any distance that is a multiple of three or four - this will be our first leg (equal to three or four parts, respectively), we get point No. 2.
For ease of calculation, you can take a distance, for example 2m (these are 4 parts of 50cm each).
Step 2. Then, from the same point No. 1, we measure 1.5 m (3 parts of 50 cm each) up (set an approximate perpendicular), draw a line (green).
Step 3. Now from point number 2 you need to put a mark on the green line at a distance of 2.5m (5 parts of 50cm). The intersection of these marks will be our point number 3.
By connecting points No. 1 and No. 3, we get a line perpendicular to our first line.
Task number 2. Second situation- there is an angle and you need to check whether it is straight.
Here it is, our corner. It's much easier to check with a large square. And if he is not?
